4. The rate at which the population of a town is increasing is 3%
per year. i.e.,

where the population, P, is measured in people and the
time, t, is measured in years.

Solve this differential equation to get the population as a
function of time. If we start measuring the time now (i.e., t
= 0 at the present) and the population of the town is 30,000 people
now, what will the population be in 5 years?

Separate the variables

Integrate both sides

Since any real number is the natural logarithm of a positive real
number, we can express our constant of integration as ln C instead of
just C. The reason we would want to do this will become apparent
shortly.

To solve for P, put both sides as powers of e

One adds exponents when one is multiplying powers of the same base

To determine the constant C, let t = 0.

In our case the population when t = o is 30,000. Our equation
becomes

To find the population in 5 years, let t = 5, and our equation
becomes

P = 30,000e((.03)(5))

= 30,000e.15

= 30,000(1.16183424)

= 34,855.027

Since we are working with people, we should round off to the
nearest whole number and predict that the population will be 34,855
people in 5 years.